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7 Plasmonics 7 Plasmonics Highlights of this chapter: In this chapter we introduce the concept of surface plasmon polaritons (SPP). We discuss various types of SPP and explain excitation methods. Finally, di®erent recent research topics and applications related to SPP are introduced. 7.1 Introduction Long before scientists have started to investigate the optical properties of metal nanostructures, they have been used by artists to generate brilliant colors in glass artefacts and artwork, where the inclusion of gold nanoparticles of di®erent size into the glass creates a multitude of colors. Famous examples are the Lycurgus cup (Roman empire, 4th century AD), which has a green color when observing in reflecting light, while it shines in red in transmitting light conditions, and church window glasses. Figure 172: Left: Lycurgus cup, right: color windows made by Marc Chagall, St. Stephans Church in Mainz Today, the electromagnetic properties of metal{dielectric interfaces undergo a steadily increasing interest in science, dating back in the works of Gustav Mie (1908) and Rufus Ritchie (1957) on small metal particles and flat surfaces. This is further moti- vated by the development of improved nano-fabrication techniques, such as electron beam lithographie or ion beam milling, and by modern characterization techniques, such as near ¯eld microscopy. Todays applications of surface plasmonics include the utilization of metal nanostructures used as nano-antennas for optical probes in biology and chemistry, the implementation of sub-wavelength waveguides, or the development of e±cient solar cells. 208 7.2 Electro-magnetics in metals and on metal surfaces 7.2.1 Basics The interaction of metals with electro-magnetic ¯elds can be completely described within the frame of classical Maxwell equations: r ¢ D = ½ (316) r ¢ B = 0 (317) r £ E = ¡@B=@t (318) r £ H = J + @D=@t; (319) which connects the macroscopic ¯elds (dielectric displacement D, electric ¯eld E, magnetic ¯eld H and magnetic induction B) with an external charge density ½ and current density J. In the limit of linear, isotropic and non-magnetic media, there are additionally the material dependent relations: D = ²0²E (320) B = ¹0H (321) with a frequency dependent dielectric constant: ² = ²(!), which is in general a 0 00 complex function, ² = ² + i²p. It is furthermore connected to the complex index of refraction via n = n + i· = ². Explicitly one can obtain the following expressions: ²0 = n2 ¡ ·2; ²00 = 2n·; (322) ²0 1p ²00 n2 = + ²02 + ²002;· = (323) 2 2 2n The real part of the refractive index n(!) is responsible for the dispersion in the medium, the imaginary part ·(!) (extinction coe±cient) determines the absorption. Beer's law describes the exponential decay of the intensity of a light beam (along the x direction) in a medium: I(x) = I0 exp(¡®x). The absorption constant can then be determined from the extinction coe±cient: ®(!) = 2·(!)!=c. 7.2.2 Bulk plasmons The optical properties can be described over a large frequency range using the plasma model, where an electron gas (e®ective electron mass M) of density N is assumed that freely propagates behind a background of positively charged atom 209 cores. These electrons start to oscillate in the presence of an electromagnetic ¯eld E(t) = E0 exp(¡i!t) and are damped through collisions with a characteristic rate γ = 1=¿ (typically ¿ ¼ 10¡14 s at room temperature). The equations of motion in this model are mxÄ + mγx_ = ¡eE(t) with the solution e x(t) = E(t) m(!2 + iγ!) The electrons which are displaced relative to the atom cores then generate a po- larisation P = ¡Nex. From this it follows for the dielectric displacement and the dielectric constant: D = ² E + P = ² ²E (324) 0µ 0 ¶ !2 = ² 1 ¡ p E (325) 0 !2 + iγ! !2 !2¿ 2 ) ²(!) = 1 ¡ p = 1 ¡ p ; (326) !2 + iγ! !2¿ 2 + i!¿ 2 2 where we introduced the plasmon frequency !p = Ne =(²0m). In case of low frequencies ·¿q¿ 1 metals are strongly absorbing. The absorption 2 2 constant then becomes ® = 2!p!¿=c . The penetration depth of the ¯elds at low frequencies after Beer's law becomes ± = 2=® and is called skin depth. However at large frequencies the approximation !¿ À 1 is valid. In this case the damping term i!¿ can be neglected and ²(!) becomes approximately real: !2 ²(!) = 1 ¡ p (327) !2 The dispersion relation of electro-magnetic ¯elds can be determined from k2 = jkj2 = 2 2 ²! =c : q 2 2 2 !(k) = !p + k =c As can be also seen in the ¯gure 173, there is no propagation of electro-magnetic waves below the plasmon frequency ! < !p. For ! > !p waves propagate with a group velocity vg = d!=dk < c. Of further interest is the special case ! = !p, where for low damping ²(!p) = 0. One can show (see e.g. S. Maier: "`Plasmonics"', p. 10 & 15f), that here a 210 2 plasmon p dispersion w / w 1 vacuum light line Frequency 0 0 1 2 Wavevector k / (w/c) p Figure 173: Dispersion relation of the free electron gas. collective longitudinal excitation mode (k k E) is formed, with a purely de-polarizing ¯eld (E = (¡1=²0)P). The physical interpretation is a collective oscillation of the conduction electron gas with respect to the ¯xed background of positive atom cores. The quanta of this charge oscillation are called plasmons (or bulk plasmons, for better discrimination with surface plasmons in the later sections). As these are longitudinal waves, bulk plasmons cannot couple to transversal electro-magnetic ¯elds and thus cannot be excited from or strayed to direct irradiation. In most metals, the plasma frequency is in the ultra-violet regime, with energies within 5-15 eV, depending on the metal band structure. 211 7.2.3 Surface plasmons Surface plasmons (or more exactly surface plasmon polaritons, SPPs) are electro- magnetic excitations that propagate along the interface between a metal and a dielectric medium. For the derivation of these excitation we again start with Maxwell's equations, which have to be separately solved for the metal and dielectric parts. Let us ¯rst start with a metal surface that extends in¯nitely in the x{y plane at z = 0 (see ¯gure 174) e dielectric 1 E1z E1 H1y z=0 E1x y x z e metal 2 Figure 174: interface along the x{y plane between a dielectric (top, index 1), and a metal (bottom, index 2). The conditions for the continuity of the normal and transversal ¯eld components on this interface (see e.g. J.D. Jackson, "`Classical Electrodynamics"') are given by: D1;z = D2z;B1;z = B2;z (328) E1;x=y = E2x=y;H1;x=y = H2;x=y (329) where the indices (1) and (2) indicate the dielectric and metal, respectively. One can show (see e.g. S. Maier: "`Plasmonics"', S. 26f), that under these condi- tions, no transverse-electric (TE) modes can exist. Instead we directly start with an ansatz for a transverse-magnetic (TM) mode for ¯eld which propagate along the 212 x direction (i = 1; 2): i(ki¢r¡i!t) Ei = (Ei;x; 0;Ei;y)e (330) i(k¢r¡i!t) Hi = (0;Hi;y; 0)e (331) Di = ²0²iEi; Bi = ¹0Hi (332) The wave vector is given by ki = (¯; 0; ki;z), where ¯ = kx indicates the propagation constant along x. Using this approach with the conditions of continuity from above and the Maxwell's equation (316) and (319) in absence of charges and currents (½ = 0, J = 0), we obtain the following relations between the ki;z components: k k 1;z = 2;z (333) ²1 ²2 We are looking for solutions that describe modes bound to the interface. Thus, the ki;z components have to be imaginary and of opposite sign: k1;z = +i·1 In this way, the ¯elds decay exponentially into the respective half spaces: Ei / exp(§iki;z) = exp(§·iz), as also symbolized in ¯gure 175. Comparing this with the relation (333), one can directly see that this is ful¯lled only, if the dielectric constants of the two materials are of opposite sign (i.e. ²1 = ¡²2). Surface plasmons can thus indeed only exist at the interface between a metal (² < 0) and a dielectric medium (² > 0). diel., e1 intensity z metal, e2 Figure 175: Evanescent ¯eld that decays exponentially into the two half spaces. Altogether, one obtains a system that is composed of an electro-magnetic wave in the dielectric medium and an oscillating electron plasma in the metal, where 213 both modes have an exponentially decaying evanescent character (see ¯gure 176). Due to this composed character, surface plasmons are also referred to as surface plasmon polaritons. The penetration depth of the ¯eld into the dielectric is typically on the order of ¸=2 of the wavelength in the medium, whereas in the metal it is characteristically given by the skin depth. Figure 176: Left: composed character of SPPs at the interface between dielectric and metal. Right: evanescent ¯elds in the two half spaces. [From Barnes et al., Nature 424, 824] Let us now derive the dispersion relations for surface plasmons. For the wave vector from eq. (330) we get: 2 2 2 2 2 2 jk1(2)j = ²1(2)k0 = ¯ + k1(2);z = ¯ ¡ ·1(2) here, k0 = !=c is the vacuum wave vector of light with frequency !. Insertion of ·1(2) into eq. (333) yields the propagation constant: r ! ² ² ¯ = 1 2 (334) c ²1 + ²2 Together with the information of the frequency dependence of ²1(2)(!), one can derive for the dispersion relation !(kSP P ). 214 Under the assumption that ²1(!) in the dielectric is approximately constant and using the approximation (327) for ²2 in the metal above the plasmon frequency, one obtains a dispersion relation as given in ¯gure 177.
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